Optimization Problems Minimum Spanning Tree Behavioral Abstraction

1
Optimization Problems Minimum Spanning
TreeBehavioral Abstraction
Lecture 17
2
Optimization Problems
3
Optimization Algorithms

• Many real-world problems involve maximizing or
  minimizing a value
• How can a car manufacturer get the most parts out
  of a piece of sheet metal?
• How can a moving company fit the most furniture
  into a truck of a certain size?
• How can the phone company route calls to get the
  best use of its lines and connections?
• How can a university schedule its classes to make
  the best use of classrooms without conflicts?

4
Optimal vs. Approximate Solutions

• Often, we can make a choice
• Do we want the guaranteed optimal solution to the
  problem, even though it might take a lot of
  work/time to find?
• Do we want to spend less time/work and find an
  approximate solution (near optimal)?

5
Approximate Solutions
6
Greedy Algorithms

• Approximates optimal solution
• May or may not find optimal solution
• Provides quick and dirty estimates
• A greedy algorithm makes a series of
  short-sighted decisions and does not look
  ahead
• Spends less time

7
A Greedy Algorithm

• Consider the weight and value of some foreign
  coins
• foo 6.00 500 grams
• bar 4.00 450 grams
• baz 3.00 410 grams
• qux 0.50 300 grams
• If we can only fit 860 grams in our pockets...
• A greedy algorithm would choose
• 1 foo 500 grams 6.00
• 1 qux 300 grams 0.50
• Optimal solution is
• 1 bar 450 grams 4.00
• 1 baz 410 grams 3.00

Total of 6.50
Total of 7.00
8
Short-Sighted Decisions
1
Start
2
End
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1
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The Shortest Path Problem

• Given a directed, acyclic, weighted graph
• Start at some vertex A
• What is the shortest path from start vertex A to
  some end vertex B?

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A Greedy Algorithm
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Start
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End
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A Greedy Algorithm
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Start
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End
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A Greedy Algorithm
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Start
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End
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A Greedy Algorithm
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Start
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End
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Path 15
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A Greedy Algorithm
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Start
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End
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Shortest Path 13
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Dynamic Planning
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Dynamic Planning

• Calculates all of the possible solution options,
  then chooses the best one.
• Implemented recursively.
• Produces an optimal solution.
• Spends more time.

17
Bellmans Principle of Optimality

• Regardless of how you reach a particular state
  (graph node), the optimal strategy for reaching
  the goal state is always the same.
• This greatly simplifies the strategy for
  searching for an optimal solution.

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The Shortest Path Problem

• Given a directed, acyclic, weighted graph
• What is the shortest path from the start vertex
  to some end vertex?
• Minimize the sum of the edge weights

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Dynamic Planning
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Start
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End
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g
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Dynamic Planning
f
Notation x means shortest path to x
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b
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End
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b min(6g, 5e, 7f)
g
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Dynamic Planning
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f
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e
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g min(6d, 14) e min(3c, 7d, 7g) f 2c
g
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Dynamic Planning
c
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Start
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d
a
c min(5, 11d) d 3
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b min(6g, 5e, 7f) g min(6d, 14) e
min(3c, 7d, 7g) f 2c c min(5, 11d) d 3
via a to d
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b min(6g, 5e, 7f) g min(6d, 14) e
min(3c, 7d, 7g) f 2c c min(5, 11d) d 3
via a to d
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b min(6g, 5e, 7f) g min(63, 14) e
min(3c, 73, 7g) f 2c c min(5, 113) d 3
via a to d
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b min(6g, 5e, 7f) g min(9, 14) e
min(3c, 10, 7g) f 2c c min(5, 14) d 3
via a to d
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b min(6g, 5e, 7f) g min(9, 14) e
min(3c, 10, 7g) f 2c c min(5, 14) d 3
via a to d
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b min(6g, 5e, 7f) g 9 via a to d to g e
min(3c, 10, 7g) f 2c c 5 via a to c d
3 via a to d
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b min(6g, 5e, 7f) g 9 via a to d to g e
min(3c, 10, 7g) f 2c c 5 via a to c d
3 via a to d
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b min(69, 5e, 7f) g 9 via a to d to g e
min(35, 10, 79) f 25 c 5 via a to c d
3 via a to d
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b min(15, 5e, 7f) g 9 via a to d to g e
min(8, 10, 16) f 7 via a to c to f c 5 via
a to c d 3 via a to d
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b min(15, 5e, 7f) g 9 via a to d to g e
min(8, 10, 16) f 7 via a to c to f c 5 via
a to c d 3 via a to d
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b min(15, 5e, 7f) g 9 via a to d to g e
8 via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
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b min(15, 5e, 7f) g 9 via a to d to g e
8 via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
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b min(15, 58, 77) g 9 via a to d to g e
8 via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
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b min(15, 13, 14) g 9 via a to d to g e 8
via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
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b min(15, 13, 14) g 9 via a to d to g e 8
via a to c to e f 7 via a to c to f c 5
via a to c d 3 via a to d
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b 13 via a to c to e to b g 9 via a to d
to g e 8 via a to c to e f 7 via a to c
to f c 5 via a to c d 3 via a to d
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Dynamic Planning
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f
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Start
11
b
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End
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Shortest Path 13
g
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Summary

• Greedy algorithms
• Make short-sighted, best guess decisions
• Required less time/work
• Provide approximate solutions
• Dynamic planning
• Examines all possible solutions
• Requires more time/work
• Guarantees optimal solution

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Questions?
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Minimum Spanning Tree

• Prims AlgorithmKruskals Algorithm

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The Scenario

• Construct a telephone network
• Weve got to connect many cities together
• Each city must be connected
• We want to minimize the total cable used

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The Scenario

• Construct a telephone network
• Weve got to connect many cities together
• Each city must be connected
• We want to minimize the total cable used

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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.

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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.

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3
2
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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights

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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights

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S.T. 3
S.T. 5
S.T. 4
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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights

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1
1
3
3
3
2
2
2
M.S.T. 3
S.T. 5
S.T. 4
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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights
• Can start at any node

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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights
• Can start at any node
• Unique solution.
• Can come from different sets of edges

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4
4
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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights
• Can start at any node
• Unique solution.
• Can come from different sets of edges

4
4
Pick any 2
4
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Minimum Spanning Tree

• Required Connect all nodes at minimum cost.
• Cost is sum of edge weights
• Can start at any node
• Unique solution.
• Can come from different sets of edges
• Two algorithms
• Prims
• Kruskals

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Prims Algorithm

• Start with any vertex.
• All its edges are candidates.
• Stop looking when all vertices are picked
• Otherwise repeat
• Pick minimum edge (no cycles) and connect the
  adjacent vertex
• Add all edges from that new vertex to our
  candidates

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Prims Algorithm

• Pick any vertex and add it to vertices list
• Loop
• Exitif (vertices contains all the vertices
  in graph)
• Select shortest, unmarked edge coming from
  vertices list
• If (shortest edge does NOT create a cycle)
  then
• Add that edge to your edges
• Add the adjoining vertex to the vertices list
• Endif
• Mark that edge as having been considered
• Endloop

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Start
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NO!
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LOOP
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Prims MST 66
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An Alternate AlgorithmKruskals Algorithm
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Kruskals Algorithm

• Sort edges in graph in increasing order
• Select the shortest edge
• Loop
• Exitif all edges examined
• Select next shortest edge (if no cycle created)
• Mark this edge as examined
• Endloop
• This guarantees an MST, but as it is built, edges
  do not have to be connected

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Sort the Edges

• 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14,
  15, 17, 18, 19, 20, 21, 24

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Still NO!
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Or
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Kruskals MST 66
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Prims MST 66
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Summary

• Minimum spanning trees
• Connect all vertices
• With no cycles
• And minimize the total edge cost
• Prims and Kruskals algorithm
• Generate minimum spanning trees
• Give same total cost, but may give different
  trees (if graph has edges with same weight)

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Questions?
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Introduction to theObject-Oriented Paradigm
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The Scenario

• Recall the concept of a Queue
• Defined by a set of behaviors
• Enqueue to the end
• Dequeue from the front
• First in, first out

Items
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Where is the Queue?

• The logical idea of the queue consisted of
• Data stored in some structure (list, array, etc.)
• Modules to act on that data
• But where is the queue?
• Wed like some way of representing sucha
  structure in our programs.

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Issues

• As is, there is no way to protect against
  violating the specified behaviors.

Enqueue
Dequeue
Sneak into Middle
Items
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Issues

• Wed like a way to put a wrapper around our
  structure to protect against this.

Enqueue
Dequeue
Sneak into Middle
Items
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Motivation

• We need ways to manage the growing complexity and
  size of programs.
• We can better model our algorithmic solutions to
  real world phenomenon.
• Contracts of responsibility can establish clearer
  communication and more protected manipulation.

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Procedural Abstraction

• Procedural Abstractions organize instructions.

Function Power Give me two numbers (base
exponent) Ill return to you baseexponent ???
Implementation ???
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Data Abstraction

• Data Abstractions organize data.

StudentType
Name (string) GPA (num) Letter Grade
(char) Student Number (num)
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Behavioral Abstraction

• Behavioral Abstractions combine procedural and
  data abstractions.

Queue Object

Enqueue
Is Full
Data State
Is Empty
Dequeue
Initialize
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The Object-Oriented Paradigm

• Instances of behavioral abstractions are known as
  objects.
• Objects have a clear interface by which they send
  and receive messages (communicate).
• OO is a design and approach issue. Just because
  a language offers object-oriented features
  doesnt mean youre doing OO programming.

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Benefits of Object-Oriented Paradigm

• Encapsulation information hiding, control, and
  design
• Reuse create libraries of tested, optimized
  classes
• Adaptability controlled interactions with
  minimal coupling
• Better model real world and larger problems
• Break down tasks better

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Information Hiding

• Information Hiding means that the user has enough
  information to use the interface, and no more

• Example stick shift
• We dont care about the inner workings
• We just want to get the car going!

1
3
5
2
4
R
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Encapsulation

• Encapsulation is the grouping of related things
  together within some structure

Item 1
Item 2
Item3
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Encapsulation via Algorithms

• Algorithms encapsulate modules, data, and
  instructions.

Algorithm
Procedure
Instructions
Function
Data
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Encapsulating Instructions

• Modules encapsulate instructions.

Procedure/Function
Instructions
Instructions
Module call
Instructions
107
Encapsulating Data

• Records allow us to do this with data

Record
Field 1
Field 2
Record
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Revisiting the Where is it? Question

• Notice we still have no way of identifying the
  idea were discussing
• The Queue is still in the ether.
• Wed like to encapsulate the data (regardless of
  its actual representation) with the behavior
  (modules).
• Once we do this, weve got a logical entity to
  which we can point and say, there it is!

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Encapsulating Data with Methods

• Abstract data types (ADTs) allow us to do this
  with logically related data and modules

Queue
Enqueue
Data
Dequeue
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Abstract Data Types

• An idea, a concept, an abstraction of the big
  picture
• It encapsulates the abstract behaviors and data
  implied by the thing being modeled.

Queue

Enqueue
Is Full
Data State
Is Empty
Dequeue
Initialize
111
Achieving Behavioral Abstraction

• Abstract data types (ADTs) are concepts.
• We require some way to implement these common
  abstractions so we can write them once, verify
  that they are correct, and reuse them.
• This would save us from having to re-do work.
  For example, every time we create a queue we did
• List_Node definesa ...q_front isoftype
  ...q_tail isoftype ...
• procedure Enqueue(...)
• procedure Dequeue(...)
• We need an algorithmic construct that will allow
  us to bundle these things together the class.

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Summary

• Behavioral abstraction combines data abstraction
  with procedural abstraction.
• The object-oriented paradigm focuses on the
  interaction and manipulation of objects.
• An Abstract Data Type (ADT) allows us to think of
  what were representing as a thing regardless of
  its actual implementation.

113
Questions?
114
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